In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative equal to its value. The exponential of a variable is denoted or , with the two notations used interchangeably. It is called exponential because its argument can be seen as an exponent to which a constant number e ≈ 2.718, the base, is raised. There are several other definitions of the exponential function, which are all equivalent although being of very different nature.
Exponential | |
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General information | |
General definition | |
Domain, codomain and image | |
Domain | |
Image | |
Specific values | |
At zero | 1 |
Value at 1 | e |
Specific features | |
Fixed point | −Wn(−1) for |
Related functions | |
Reciprocal | |
Inverse | Natural logarithm, Complex logarithm |
Derivative | |
Antiderivative | |
Series definition | |
Taylor series |
The exponential function converts sums to products: it maps the additive identity 0 to the multiplicative identity 1, and the exponential of a sum is equal to the product of separate exponentials, . Its inverse function, the natural logarithm, or , converts products to sums: .
Other functions of the general form , with base , are also commonly called exponential functions, and share the property of converting addition to multiplication, . Where these two meanings might be confused, the exponential function of base is occasionally called the natural exponential function, matching the name natural logarithm. The generalization of the standard exponent notation to arbitrary real numbers as exponents, is usually formally defined in terms of the exponential and natural logarithm functions, as . The "natural" base is the unique base satisfying the criterion that the exponential function's derivative equals its value, , which simplifies definitions and eliminates extraneous constants when using exponential functions in calculus.
Quantities which change over time in proportion to their value, for example the balance of a bank account bearing compound interest, the size of a bacterial population, the temperature of an object relative to its environment, or the amount of a radioactive substance, can be modeled using functions of the form , also sometimes called exponential functions; these quantities undergo exponential growth if is positive or exponential decay if is negative.
The exponential function can be generalized to accept a complex number as its argument. This reveals a relation between the multiplication of complex numbers and rotation in the Euclidean plane, Euler's formula : the exponential of an imaginary number is a point on the complex unit circle at angle from the real axis. The identities of trigonometry can thus be translated into identities involving exponentials of imaginary quantities. The complex function is a conformal map from an infinite strip of the complex plane (which periodically repeats in the imaginary direction) onto the whole complex plane except for .
The exponential function can be even further generalized to accept other types of arguments, such as matrices and elements of Lie algebras.
Graph
editThe graph of is upward-sloping, and increases faster as x increases.[1] The graph always lies above the x-axis, but becomes arbitrarily close to it for large negative x; thus, the x-axis is a horizontal asymptote. The equation means that the slope of the tangent to the graph at each point is equal to its height (its y-coordinate) at that point.
Definitions and fundamental properties
editThere are several different definitions of the exponential function, which are all equivalent, although of very different nature.
One of the simplest definitions is: The exponential function is the unique differentiable function that equals its derivative, and takes the value 1 for the value 0 of its variable.
This "conceptual" definition requires a uniqueness proof and an existence proof, but it allows an easy derivation of the main properties of the exponential function.
Uniqueness: If and are two functions satisfying the above definition, then the derivative of is zero everywhere by the quotient rule. It follows that is constant, and this constant is 1 since .
The exponential function is the inverse function of the natural logarithm. The inverse function theorem implies that the natural logarithm has an inverse function, that satisfies the above definition. This is a first proof of existence. Therefore, one has
for every real number and every positive real number
The exponential function is the sum of a power series:[2][3] where is the factorial of n (the product of the n first positive integers). This series is absolutely convergent for every per the ratio test. So, the derivative of the sum can be computed by term-by-term derivation, and this shows that the sum of the series satisfies the above definition. This is a second existence proof, and shows, as a byproduct, that the exponential function is defined for every , and is everywhere the sum of its Maclaurin series.
The exponential satisfies the functional equation: This results from the uniqueness and the fact that the function satisfies the above definition. It can be proved that a function that satisfies this functional equation is the exponential function if its derivative at 0 is 1 and the function is either continuous or monotonic
Positiveness: For every , one has , since the functional equation implies . It results that the exponential function is positive (since , if one would have for some , the intermediate value theorem would imply the existence of some such that . It results also that the exponential function is monotonically increasing.
Extension of exponentiation to positive real bases: Let b be a positive real number. The exponential function and the natural logarithm being the inverse each of the other, one has If n is an integer, the functional equation of the logarithm implies Since the right-most expression is defined if n is any real number, this allows defining for every positive real number b and every real number x: In particular, if b is the Euler's number one has (inverse function) and thus This shows the equivalence of the two notations for the exponential function.
The exponential function is the limit[4][3] where takes only integer values (otherwise, the exponentiation would require the exponential function to be defined). By continuity of the logarithm, this can be proved by taking logarithms and proving for example with Taylor's theorem.
General exponential functions
editSometimes 'exponential function' is used as the name for all functions with the argument somewhere in the exponent of an exponentiation: as well as ;[5] this has serious disadvantages.[6] In this section the name 'exponential function' will be used for functions obeying one of the following (equivalent) conditions:
- For all , , : (pairs of arguments with the same difference in the domain, are mapped into pairs of values with the same ratio in the codomain). Or: for all , the value of is independent of .[7][8][9][10]
- For all , : . Or: the value of is independent of . This constant value is sometimes called the rate constant of function , symbol .[11][12][13]
- The value of is independent of and . Or: the value of is independent of and . This constant value is called the base of function .[14]
Expressions for exponential functions
editThe value of argument (real or complex) of an exponential function can be expressed (with , , ) as :
-
- ( , )
-
-
Exponential functions mapping quantities instead of numbers,are usually expressed as ; quantity being the unit in which quantity (mostly 'time') is measured, resulting in a dimensionless exponent.
Hierarchy of types
edit- Exponential functions with quantities as elements of domain and codomain. E.g. the lilys in the pond, growing with the same factor during time intervals of equal length. In applications in empirical sciences, notations with and are commonly used. Exponential growth can be modeled by a function with its doubling time. Exponential decay can be modeled by a function with its half-life.
- Exponential functions with domain ; see § Complex exponential, below.
- Exponential functions obeying for all , (changing additions into multiplications; the opposite of the main property of logarithmic functions: changing multiplications into additions) ; equivalent with the condition . Usual form:
Sometimes the value of is named the antilog of or the antilogarithm of .[15] - Exponential functions obeying (the function is identical with its own derivative). Usual form:
- The (unique) exponential function obeying as well as is called the exponential function; sometimes the natural exponential function or the natural antilogarithm. Symbol: . Usual form: or
Two meanings of 'base'
editFor exponential functions , to , the -independent value of is called the base of the function .[16] While in expressions (...)(...) and (...)^(...) the value of the first element is called the base of the exponentiation. Example: the exponential function has base , while the expression has base (and exponent ).
Properties
edit- The Euler number is connected with every exponential function . When argument increases by , changes by factor . For .
- The graph of an exponential function in polar coordinates is a logarithmic spiral or equiangular spiral.[17][18] In a logarithmic spiral with pitch angle 45o the length of a radius vector increases by a factor when the polar angle increases by one radian. And by the factor at a 180o switch. See , logarithmic spiral, §Properties, 'Rotating, scaling'.
- An exponential function is determined by two 'points'. With , positive, and determine the exponential function .[19]
Overview
editThe exponential function arises whenever a quantity grows or decays at a rate proportional to its current value. One such situation is continuously compounded interest, and in fact it was this observation that led Jacob Bernoulli in 1683[20] to the number now known as e. Later, in 1697, Johann Bernoulli studied the calculus of the exponential function.[20]
If a principal amount of 1 earns interest at an annual rate of x compounded monthly, then the interest earned each month is x/12 times the current value, so each month the total value is multiplied by (1 + x/12), and the value at the end of the year is (1 + x/12)12. If instead interest is compounded daily, this becomes (1 + x/365)365. Letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function, first given by Leonhard Euler.[4] This is one of a number of characterizations of the exponential function; others involve series or differential equations.
From any of these definitions it can be shown that e−x is the reciprocal of ex. For example, from the differential equation definition, ex e−x = 1 when x = 0 and its derivative using the product rule is ex e−x − ex e−x = 0 for all x, so ex e−x = 1 for all x.
From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity. For example, from the power series definition, expanded by the Binomial theorem, This justifies the exponential notation ex for exp x.
The derivative (rate of change) of the exponential function is the exponential function itself. More generally, a function with a rate of change proportional to the function itself is expressible in terms of the exponential function. This derivative property leads to exponential growth or exponential decay.
The exponential function extends to an entire function on the complex plane. Euler's formula relates its values at purely imaginary arguments to trigonometric functions. The exponential function also has analogues for which the argument is a matrix, or even an element of a Banach algebra or a Lie algebra.
Derivatives and differential equations
editThe importance of the exponential function in mathematics and the sciences stems mainly from its property as the unique function which is equal to its derivative and is equal to 1 when x = 0. That is,
Functions of the form aex for constant a are the only functions that are equal to their derivative (by the Picard–Lindelöf theorem). Other ways of saying the same thing include:
- The slope of the graph at any point is the height of the function at that point.
- The rate of increase of the function at x is equal to the value of the function at x.
- The function solves the differential equation y′ = y.
- exp is a fixed point of derivative as a linear operator on function space.
If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth (see Malthusian catastrophe), continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time.
More generally, for any real constant k, a function f: R → R satisfies if and only if for some constant a. The constant k is called the decay constant, disintegration constant,[21] rate constant,[22] or transformation constant.[23]
Furthermore, for any differentiable function f, we find, by the chain rule:
Continued fractions for ex
editA continued fraction for ex can be obtained via an identity of Euler:
The following generalized continued fraction for ez converges more quickly:[24]
or, by applying the substitution z = x/y: with a special case for z = 2:
This formula also converges, though more slowly, for z > 2. For example:
Complex exponential
editAs in the real case, the exponential function can be defined on the complex plane in several equivalent forms.
The most common definition of the complex exponential function parallels the power series definition for real arguments, where the real variable is replaced by a complex one:
Alternatively, the complex exponential function may be defined by modelling the limit definition for real arguments, but with the real variable replaced by a complex one:
For the power series definition, term-wise multiplication of two copies of this power series in the Cauchy sense, permitted by Mertens' theorem, shows that the defining multiplicative property of exponential functions continues to hold for all complex arguments:
The definition of the complex exponential function in turn leads to the appropriate definitions extending the trigonometric functions to complex arguments.
In particular, when z = it (t real), the series definition yields the expansion
In this expansion, the rearrangement of the terms into real and imaginary parts is justified by the absolute convergence of the series. The real and imaginary parts of the above expression in fact correspond to the series expansions of cos t and sin t, respectively.
This correspondence provides motivation for defining cosine and sine for all complex arguments in terms of and the equivalent power series:[25]
for all
The functions exp, cos, and sin so defined have infinite radii of convergence by the ratio test and are therefore entire functions (that is, holomorphic on ). The range of the exponential function is , while the ranges of the complex sine and cosine functions are both in its entirety, in accord with Picard's theorem, which asserts that the range of a nonconstant entire function is either all of , or excluding one lacunary value.
These definitions for the exponential and trigonometric functions lead trivially to Euler's formula:
We could alternatively define the complex exponential function based on this relationship. If z = x + iy, where x and y are both real, then we could define its exponential as where exp, cos, and sin on the right-hand side of the definition sign are to be interpreted as functions of a real variable, previously defined by other means.[26]
For , the relationship holds, so that for real and maps the real line (mod 2π) to the unit circle in the complex plane. Moreover, going from to , the curve defined by traces a segment of the unit circle of length starting from z = 1 in the complex plane and going counterclockwise. Based on these observations and the fact that the measure of an angle in radians is the arc length on the unit circle subtended by the angle, it is easy to see that, restricted to real arguments, the sine and cosine functions as defined above coincide with the sine and cosine functions as introduced in elementary mathematics via geometric notions.
The complex exponential function is periodic with period 2πi and holds for all .
When its domain is extended from the real line to the complex plane, the exponential function retains the following properties:
for all
Extending the natural logarithm to complex arguments yields the complex logarithm log z, which is a multivalued function.
We can then define a more general exponentiation: for all complex numbers z and w. This is also a multivalued function, even when z is real. This distinction is problematic, as the multivalued functions log z and zw are easily confused with their single-valued equivalents when substituting a real number for z. The rule about multiplying exponents for the case of positive real numbers must be modified in a multivalued context:
≠ ezw, but rather (ez)w
= e(z + 2niπ)w multivalued over integers n
See failure of power and logarithm identities for more about problems with combining powers.
The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin. Two special cases exist: when the original line is parallel to the real axis, the resulting spiral never closes in on itself; when the original line is parallel to the imaginary axis, the resulting spiral is a circle of some radius.
-
z = Re(ex + iy)
-
z = Im(ex + iy)
-
z = |ex + iy|
Considering the complex exponential function as a function involving four real variables: the graph of the exponential function is a two-dimensional surface curving through four dimensions.
Starting with a color-coded portion of the domain, the following are depictions of the graph as variously projected into two or three dimensions.
-
Checker board key:
-
Projection onto the range complex plane (V/W). Compare to the next, perspective picture.
-
Projection into the , , and dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image)
-
Projection into the , , and dimensions, producing a spiral shape ( range extended to ±2π, again as 2-D perspective image)
The second image shows how the domain complex plane is mapped into the range complex plane:
- zero is mapped to 1
- the real axis is mapped to the positive real axis
- the imaginary axis is wrapped around the unit circle at a constant angular rate
- values with negative real parts are mapped inside the unit circle
- values with positive real parts are mapped outside of the unit circle
- values with a constant real part are mapped to circles centered at zero
- values with a constant imaginary part are mapped to rays extending from zero
The third and fourth images show how the graph in the second image extends into one of the other two dimensions not shown in the second image.
The third image shows the graph extended along the real axis. It shows the graph is a surface of revolution about the axis of the graph of the real exponential function, producing a horn or funnel shape.
The fourth image shows the graph extended along the imaginary axis. It shows that the graph's surface for positive and negative values doesn't really meet along the negative real axis, but instead forms a spiral surface about the axis. Because its values have been extended to ±2π, this image also better depicts the 2π periodicity in the imaginary value.
Matrices and Banach algebras
editThe power series definition of the exponential function makes sense for square matrices (for which the function is called the matrix exponential) and more generally in any unital Banach algebra B. In this setting, e0 = 1, and ex is invertible with inverse e−x for any x in B. If xy = yx, then ex + y = exey, but this identity can fail for noncommuting x and y.
Some alternative definitions lead to the same function. For instance, ex can be defined as
Or ex can be defined as fx(1), where fx : R → B is the solution to the differential equation dfx/dt(t) = x fx(t), with initial condition fx(0) = 1; it follows that fx(t) = etx for every t in R.
Lie algebras
editGiven a Lie group G and its associated Lie algebra , the exponential map is a map ↦ G satisfying similar properties. In fact, since R is the Lie algebra of the Lie group of all positive real numbers under multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie group GL(n,R) of invertible n × n matrices has as Lie algebra M(n,R), the space of all n × n matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.
The identity can fail for Lie algebra elements x and y that do not commute; the Baker–Campbell–Hausdorff formula supplies the necessary correction terms.
Transcendency
editThe function ez is not in the rational function ring : it is not the quotient of two polynomials with complex coefficients.
If a1, ..., an are distinct complex numbers, then ea1z, ..., eanz are linearly independent over , and hence ez is transcendental over .
Computation
editThe Taylor series definition above is generally efficient for computing (an approximation of) . However, when computing near the argument , the result will be close to 1, and computing the value of the difference with floating-point arithmetic may lead to the loss of (possibly all) significant figures, producing a large relative error, possibly even a meaningless result.
Following a proposal by William Kahan, it may thus be useful to have a dedicated routine, often called expm1
, which computes ex − 1 directly, bypassing computation of ex. For example,
one may use the Taylor series:
This was first implemented in 1979 in the Hewlett-Packard HP-41C calculator, and provided by several calculators,[27][28] operating systems (for example Berkeley UNIX 4.3BSD[29]), computer algebra systems, and programming languages (for example C99).[30]
In addition to base e, the IEEE 754-2008 standard defines similar exponential functions near 0 for base 2 and 10: and .
A similar approach has been used for the logarithm (see lnp1).[nb 1]
An identity in terms of the hyperbolic tangent, gives a high-precision value for small values of x on systems that do not implement expm1(x).
See also
edit- Carlitz exponential, a characteristic p analogue
- Double exponential function – Exponential function of an exponential function
- Exponential field – Mathematical field with an extra operation
- Gaussian function
- Half-exponential function, a compositional square root of an exponential function
- Lambert W function#Solving equations – Multivalued function in mathematics - Used for solving exponential equations
- List of exponential topics
- List of integrals of exponential functions
- Mittag-Leffler function, a generalization of the exponential function
- p-adic exponential function
- Padé table for exponential function – Padé approximation of exponential function by a fraction of polynomial functions
- Phase factor
Notes
edit- ^ A similar approach to reduce round-off errors of calculations for certain input values of trigonometric functions consists of using the less common trigonometric functions versine, vercosine, coversine, covercosine, haversine, havercosine, hacoversine, hacovercosine, exsecant and excosecant.
References
edit- ^ "Exponential Function Reference". www.mathsisfun.com. Retrieved 2020-08-28.
- ^ Rudin, Walter (1987). Real and complex analysis (3rd ed.). New York: McGraw-Hill. p. 1. ISBN 978-0-07-054234-1.
- ^ a b Weisstein, Eric W. "Exponential Function". mathworld.wolfram.com. Retrieved 2020-08-28.
- ^ a b Maor, Eli. e: the Story of a Number. p. 156.
- ^ H.A. Lorentz, Lehrbuch der Differential- und Integralrechnung, 1. Auflage 1900, S. 15 [1]; 3. Auflage 1915, S. 44;
"Funktionen, bei denen die unabhänglige Variabele im Exponenten einer Potenz auftritt, wie zum Beispiel , , , nennt man exponentiellen Funktionen". (Functions with the independent variable occurring in the exponent of an exponentiation, are called exponential functions, e. g. ...). - ^ (1) Every (positive) function f is exponential, for it can be written as .
(2) The function is not exponential. - ^ G. Harnett, Calculus 1, 1998, Functions continued "General exponential functions have the property that the ratio of two outputs depends only on the difference of inputs. The ratio of outputs for a unit change in input is the base."
- ^ G. Harnett, Quora, 2020, What is the base of an exponential function?
"A (general) exponential function changes by the same factor over equal increments of the input. The factor of change over a unit increment is called the base." - ^ Kansas State University [2]
"What makes exponential functions unique, is that outputs at inputs with constant difference have the same ratio." - ^ Mathebibel [3]
"Werden bei einer Exponentialfunktion zur basis die -Werte jeweils um einen festen Zahlenwert vergrössert, so werden die Funktionswerte mit einem konstanten Faktor vervielfacht." - ^ H. Lamb, An Elementary Course of Infinitesimal Calculus, 3rd ed. 1919 (reprint 1927), p. 72 [4] "their fundamental property is that [..] the rate of increase bears always a constant ratio to the instantaneous value of the function."
- ^ G.F. Simmons, Differential Equations and Historical Notes, 1st ed. 1972, p. 15; 3rd ed. 2016, p. 23
"The positive constant is called the rate constant, for its value is clearly a measure of the rate at which the reaction proceeds." [5]. - ^ Worcester Polytechnic Institute, Exponential growth and decay
- ^ This defining condition is derivable from the usual way to describe an exponential function: with independent of (divide both sides by , exponentiate with , replace with the more general , and replace variables , with , ) .
- ^ F. Durell, Plane and Spherical Trigonometry, 1911, p. 12.
"Inverse Use of a Table of Logarithms; that is, given a logarithm, to find the number corresponding to it (called its antilogarithm)." - ^ G. Harnett, Calculus 1, 1998; Functions continued / Exponentials & logarithms
"The ratio of outputs for a unit change in input is the base of a general exponential function." - ^ D. Hemenway, Divine Proportion, 2005, p. 127.
- ^ Ch.-J. de la Vallée Poussin, Cours d'Analyse Infinitésimale, Tome I, 3ième édition 1914, p. 363 [6]
- ^ Stack Exchange, 2019
- ^ a b O'Connor, John J.; Robertson, Edmund F. (September 2001). "The number e". School of Mathematics and Statistics. University of St Andrews, Scotland. Retrieved 2011-06-13.
- ^ Serway, Raymond A.; Moses, Clement J.; Moyer, Curt A. (1989). Modern Physics. Fort Worth: Harcourt Brace Jovanovich. p. 384. ISBN 0-03-004844-3.
- ^ Simmons, George F. (1972). Differential Equations with Applications and Historical Notes. New York: McGraw-Hill. p. 15. LCCN 75173716.
- ^ McGraw-Hill Encyclopedia of Science & Technology (10th ed.). New York: McGraw-Hill. 2007. ISBN 978-0-07-144143-8.
- ^ Lorentzen, L.; Waadeland, H. (2008). "A.2.2 The exponential function.". Continued Fractions. Atlantis Studies in Mathematics. Vol. 1. p. 268. doi:10.2991/978-94-91216-37-4. ISBN 978-94-91216-37-4.
- ^ Rudin, Walter (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. p. 182. ISBN 978-0-07-054235-8.
- ^ Apostol, Tom M. (1974). Mathematical Analysis (2nd ed.). Reading, Mass.: Addison Wesley. pp. 19. ISBN 978-0-201-00288-1.
- ^ HP 48G Series – Advanced User's Reference Manual (AUR) (4 ed.). Hewlett-Packard. December 1994 [1993]. HP 00048-90136, 0-88698-01574-2. Retrieved 2015-09-06.
- ^ HP 50g / 49g+ / 48gII graphing calculator advanced user's reference manual (AUR) (2 ed.). Hewlett-Packard. 2009-07-14 [2005]. HP F2228-90010. Retrieved 2015-10-10. [7]
- ^ Beebe, Nelson H. F. (2017-08-22). "Chapter 10.2. Exponential near zero". The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library (1 ed.). Salt Lake City, UT, USA: Springer International Publishing AG. pp. 273–282. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. LCCN 2017947446. S2CID 30244721.
Berkeley UNIX 4.3BSD introduced the expm1() function in 1987.
- ^ Beebe, Nelson H. F. (2002-07-09). "Computation of expm1 = exp(x)−1" (PDF). 1.00. Salt Lake City, Utah, USA: Department of Mathematics, Center for Scientific Computing, University of Utah. Retrieved 2015-11-02.
External links
edit- "Exponential function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]